Estamos trabajando para incorporar este artículo al repositorio
Consulte el artículo en la página del editor
Consulte la política de Acceso Abierto del editor


Several equivariant estimators of multivariate location and scatter are studied, which are highly robust, have a controllable finite-sample efficiency and are computationally feasible in large dimensions. The most frequently employed estimators are not quite satisfactory in this respect. The Minimum Volume Ellipsoid (MVE) and the Minimum Covariance Determinant (MCD) estimators are known to have a very low efficiency. S-estimators with a monotonic weight function like the bisquare have a low efficiency when the dimension p is small, and their efficiency tends to one with increasing p. Unfortunately, this advantage is outweighed by a serious loss in robustness for large p. Four families of estimators with controllable efficiencies whose performance for moderate to large p has not been explored to date are studied: S-estimators with a non-monotonic weight function, MM-estimators, τ-estimators, and the Stahel–Donoho estimator. Two types of starting estimators are employed: the MVE computed through subsampling, and a semi-deterministic procedure previously proposed for outlier detection, based on the projections with maximum and minimum kurtosis. A simulation study shows that an S-estimator with non-monotonic weight function can simultaneously attain high efficiency and high robustness for p≥15, while an MM-estimator with a particular weight function can be recommended for p>15. For both recommended estimators, the initial values are given by the semi-deterministic procedure mentioned above. © 2016 Elsevier B.V.


Documento: Artículo
Título:Robust and efficient estimation of multivariate scatter and location
Autor:Maronna, R.A.; Yohai, V.J.
Filiación:Department of Mathematics, Facultad de Ciencias Exactas, Universidad Nacional de La Plata, Argentina
Department of Mathematics, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and CONICET, Argentina
Palabras clave:Kullback–Leibler divergence; MM-estimator; S-estimator; Stahel–Donoho estimator; τ-estimator; Multivariable systems; Sampling; Efficient estimation; Equivariant estimators; Minimum covariance determinant; Minimum volume ellipsoids; MM-estimator; Outlier Detection; S-estimators; Simulation studies; Efficiency
Página de inicio:64
Página de fin:75
Título revista:Computational Statistics and Data Analysis
Título revista abreviado:Comput. Stat. Data Anal.


  • Agostinelli, C., Leung, A., Yohai, V.J., Zamar, R.H., Robust estimation of multivariate location and scatter in the presence of cellwise and casewise contamination (2015) TEST, 24, pp. 441-461
  • Croux, C., Haesbroeck, G., Influence function and efficiency of the minimum covariance determinant scatter matrix estimator (1999) J. Multivariate Anal., 71, pp. 161-190
  • Davies, P.L., Asymptotic behaviour of S-estimates of multivariate location parameters and dispersion matrices (1987) Ann. Statist., 15, pp. 1269-1292
  • Donoho, D., Breakdown properties of multivariate location estimators (1982), (Ph.D. thesis) Harvard University unpublished thesis; Gnanadesikan, R., Kettenring, J.R., Robust estimates, residuals, and outlier detection with multiresponse data (1972) Biometrics, 28, pp. 81-124
  • Hampel, F.R., Ronchetti, E.M., Rousseeuw, P.J., Stahel, W.A., Robust Statistics: The Approach Based on Influence Functions (1986), John Wiley & Sons; Hubert, M., Rousseeuw, P., Vanpaemel, D., Verdonck, T., The DetS and DetMM estimators for multivariate location and scatter (2015) Data Anal., 81, pp. 64-75
  • Janssens, K.H., Deraedt, I., Schalm, O., Veeckman, J., Composition of 15–17th Century Archaeological Glass Vessels Excavated in Antwerp, Belgium (1998), pp. 253-267. , Springer Vienna Vienna; Kent, J.T., Tyler, D.E., Constrained M-estimation for multivariate location and scatter (1996) Ann. Statist., 24, pp. 1346-1370
  • Locantore, N., Marron, J., Simpson, D., Tripoli, N., Zhang, J., Cohen, K., Robust principal component analysis for functional data (1999) TEST, 8, pp. 1-73
  • Lopuhaä, H.P., Multivariate τ-estimators for location and scatter (1991) Canad. J. Statist., 19, pp. 307-321
  • Lopuhaä, H.P., Highly efficient estimators of multivariate location with high breakdown point (1992) Ann. Statist., 20, pp. 398-413
  • Maronna, R.A., Robust M-estimators of multivariate location and scatter (1976) Ann. Statist., 4, pp. 51-67
  • Maronna, R.A., Martin, D.R., Yohai, V.J., Robust Statistics: Theory and Methods (2006), Wiley; Maronna, R.A., Yohai, V.J., The behavior of the Stahel–Donoho robust multivariate estimator (1995) J. Amer. Statist. Assoc., 90, pp. 330-341
  • Muler, N., Yohai, V., Robust estimates for ARCH processes (2002) J. Time Series Anal., 23, pp. 341-375
  • Paindaveine, D., Bever, G.V., Inference on the shape of elliptical distributions based on the MCD (2014) J. Multivariate Anal., 129, pp. 125-144
  • Peña, D., Prieto, F., Combining random and specific directions for outlier detection and robust estimation in high-dimensional multivariate data (2007) J. Comput. Graph. Statist., 16, pp. 228-254
  • Rocke, D.M., Robustness properties of S-estimators of multivariate location and shape in high dimension (1996) Ann. Statist., 24, pp. 1327-1345
  • Rousseeuw, P., Multivariate Estimation with High Breakdown Point (1985), pp. 283-297. , Reidel Publishing Company Dordrecht; Stahel, W., Breakdown of covariance estimators. Tech. Rep. (1981), E.T.H. Zürich; Tatsuoka, K.S., Tyler, D.E., On the uniqueness of S-functionals and M-functionals under nonelliptical distributions (2000) Ann. Statist., 28, pp. 1219-1243
  • Tyler, D.E., Finite sample breakdown points of projection based multivariate location and scatter statistics (1994) Ann. Statist., 22, pp. 1024-1044
  • Verboven, S., Hubert, M., Libra: a MATLAB library for robust analysis (2005) Chemometr. Intell. Lab. Syst., 75, pp. 127-136
  • Yohai, V.J., High breakdown-point and high efficiency robust estimates for regression (1987) Ann. Statist., 15, pp. 642-656
  • Yohai, V.J., Zamar, R.H., High breakdown-point estimates of regression by means of the minimization of an efficient scale (1988) J. Amer. Statist. Assoc., 83, pp. 406-413
  • Zuo, Y., Cui, H., He, X., On the Stahel–Donoho estimator and depth-weighted means of multivariate data (2004) Ann. Statist., 32, pp. 167-188


---------- APA ----------
Maronna, R.A. & Yohai, V.J. (2017) . Robust and efficient estimation of multivariate scatter and location. Computational Statistics and Data Analysis, 109, 64-75.
---------- CHICAGO ----------
Maronna, R.A., Yohai, V.J. "Robust and efficient estimation of multivariate scatter and location" . Computational Statistics and Data Analysis 109 (2017) : 64-75.
---------- MLA ----------
Maronna, R.A., Yohai, V.J. "Robust and efficient estimation of multivariate scatter and location" . Computational Statistics and Data Analysis, vol. 109, 2017, pp. 64-75.
---------- VANCOUVER ----------
Maronna, R.A., Yohai, V.J. Robust and efficient estimation of multivariate scatter and location. Comput. Stat. Data Anal. 2017;109:64-75.